Abstract
Earlier, I.V. Latkin and the author have shown the set partition problem can be reduced to the problem of finding singular points of a cubic hypersurface. The article focuses on the new link between two different research areas as well as on methods to look for singular points or to confirm the smoothness of the hypersurface. Our approach is based on the description of tangent lines to the hypersurface. The existence of at least one singular point imposes a restriction on the algebraic equation that determines the set of tangent lines passing through the selected point of the space. This equation is based on the formula for the discriminant of a univariate polynomial. We have proposed a probabilistic algorithm for some set of inputs of the set partition problem. The probabilistic algorithm is not proved to have polynomial complexity.
The research has been carried out at the expense of the Russian Science Foundation, project no. 14–50–00150.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)
Margulies, S., Onn, S., Pasechnik, D.V.: On the complexity of Hilbert refutations for partition. J. Symbolic Comput. 66, 70–83 (2015). doi:10.1016/j.jsc.2013.06.005
Herrero, M.I., Jeronimo, G., Sabia, J.: Affine solution sets of sparse polynomial systems. J. Symbolic Comput. 51, 34–54 (2013). doi:10.1016/j.jsc.2012.03.006
Bodur, M., Dash, S., Günlük, O.: Cutting planes from extended LP formulations. Math. Program. 161(1), 159–192 (2017). doi:10.1007/s10107-016-1005-7
Tamir, A.: New pseudopolynomial complexity bounds for the bounded and other integer Knapsack related problems. Oper. Res. Lett. 37(5), 303–306 (2009). doi:10.1016/j.orl.2009.05.003
Claßen, G., Koster, A.M.C.A., Schmeink, A.: The multi-band robust knapsack problem — a dynamic programming approach. Discrete Optimization. 18, 123–149 (2015). doi:10.1016/j.disopt.2015.09.007
Schwartz, J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27(4), 701–717 (1980). doi:10.1145/322217.322225
Kapovich, I., Myasnikov, A., Schupp, P., Shpilrain, V.: Generic-case complexity, decision problems in group theory, and random walks. J. Algebra 264, 665–694 (2003). doi:10.1016/S0021-8693(03)00167-4
Rybalov, A.N.: A generic relation on recursively enumerable sets. Algebra Logic 55(5), 387–393 (2016). doi:10.1007/s10469-016-9410-9
Latkin, I.V., Seliverstov, A.V.: Computational complexity of fragments of the theory of complex numbers. Bulletin of University of Karaganda. Ser. Mathematics, vol. 1, pp. 47–55 (2015). (in Russian)
Seliverstov, A.V.: On cubic hypersurfaces with involutions. In: Vassiliev, N.N. (ed.) International Conference Polynomial Computer Algebra 2016, St. Petersburg, 18–22 April 2016, pp. 74–77. VVM Publishing, Saint Petersburg (2016)
Nesterov, Y.: Random walk in a simplex and quadratic optimization over convex polytopes. CORE Discussion Paper 2003/71 (2003)
Hillar, C.J., Lim, L.H.: Most tensor problems are NP-hard. J. ACM 60(6), 45 (2013). doi:10.1145/2512329
Gel’fand, I.M., Zelevinskii, A.V., Kapranov, M.M.: Discriminants of polynomials in several variables and triangulations of Newton polyhedra. Leningrad Math. J. 2(3), 499–505 (1991)
Chistov, A.L.: An improvement of the complexity bound for solving systems of polynomial equations. J. Math. Sci. 181(6), 921–924 (2012). doi:10.1007/s10958-012-0724-4
Kulikov, V.R., Stepanenko, V.A.: On solutions and Waring’s formulae for the system of \(n\) algebraic equations with \(n\) unknowns. St. Petersburg Math. J. 26(5), 839–848 (2015). doi:10.1090/spmj/1361
Bokut, L.A., Chen, Y.: Gröbner-Shirshov bases and their calculation. Bull. Math. Sci. 4(3), 325–395 (2014). doi:10.1007/s13373-014-0054-6
Bardet, M., Faugère, J.-C., Salvy, B.: On the complexity of the \(F_5\) Gröbner basis algorithm. J. Symbolic Comput. 70, 49–70 (2015). doi:10.1016/j.jsc.2014.09.025
Eder, C., Faugère, J.-C.: A survey on signature-based algorithms for computing Gröbner bases. J. Symbolic Comput. 80(3), 719–784 (2017). doi:10.1016/j.jsc.2016.07.031
Mayr, E.W., Ritscher, S.: Dimension-dependent bounds for Gröbner bases of polynomial ideals. J. Symbolic Comput. 49, 78–94 (2013). doi:10.1016/j.jsc.2011.12.018
Malaschonok, G., Scherbinin, A.: Triangular decomposition of matrices in a domain. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2015. LNCS, vol. 9301, pp. 292–306. Springer, Cham (2015). doi:10.1007/978-3-319-24021-3_22
Vershik, A.M., Sporyshev, P.V.: An estimate of the average number of steps in the simplex method, and problems in asymptotic integral geometry. Sov. Math. Dokl. 28, 195–199 (1983)
Smale, S.: On the average number of steps of the simplex method of linear programming. Math. Program. 27(3), 241–262 (1983). doi:10.1007/BF02591902
Dubickas, A., Smyth, C.J.: Length of the sum and product of algebraic numbers. Math. Notes. 77, 787–793 (2005). doi:10.1007/s11006-005-0079-y
Kollár, J.: Unirationality of cubic hypersurfaces. J. Inst. Math. Jussieu. 1(3), 467–476 (2002). doi:10.1017/S1474748002000117
Cenk, M., Hasan, M.A.: On the arithmetic complexity of Strassen-like matrix multiplications. J. Symbolic Comput. 80(2), 484–501 (2017). doi:10.1016/j.jsc.2016.07.004
Hedén, I.: Russell’s hypersurface from a geometric point of view. Osaka J. Math. 53(3), 637–644 (2016)
Acknowledgements
The author would like to thank Mark Spivakovsky, Sergei P. Tarasov, Mikhail N. Vyalyi, and the anonymous reviewers for useful comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Seliverstov, A.V. (2017). On Probabilistic Algorithm for Solving Almost All Instances of the Set Partition Problem. In: Weil, P. (eds) Computer Science – Theory and Applications. CSR 2017. Lecture Notes in Computer Science(), vol 10304. Springer, Cham. https://doi.org/10.1007/978-3-319-58747-9_25
Download citation
DOI: https://doi.org/10.1007/978-3-319-58747-9_25
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-58746-2
Online ISBN: 978-3-319-58747-9
eBook Packages: Computer ScienceComputer Science (R0)